On the spectrum of genera of quotients of the Hermitian curve

Abstract

We investigate the genera of quotient curves ${\mathcal{H}}_q/G$ of the ${\mathbb{F}}_{q^2}$-maximal Hermitian curve ${\mathcal{H}}_q$, where $G$ is contained in the maximal subgroup ${\mathcal{M}}_q\le \mathrm{Aut}({\mathcal{H}}_q)$ fixing a pole-polar pair $(P,\ell )$ with respect to the unitary polarity associated with ${\mathcal{H}}_q$. To this aim, a geometric and group-theoretical description of ${\mathcal{M}}_q$ is given. The genera of some other quotients ${\mathcal{H}}_q/G$ with $G\nleqq {\mathcal{M}}_q$ are also computed. Thus we obtain new values in the spectrum of genera of ${\mathbb{F}}_{q^2}$-maximal curves. A plane model for ${\mathcal{H}}_q/G$ is obtained when $G$ is cyclic of order $p\cdot d$, with $d$ a divisor of $q+1$.

References in corpus (1)

• A complete characterization of Galois subfields of the generalized Giulietti--Korchmáros function field